Problem: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}2x+9y &= 1 \\ -6x-9y &= 5\end{align*}$
Begin by moving the $y$ -term in the second equation to the right side of the equation. $-6x = 9y+5$ Divide both sides by $-6$ to isolate $x$ $x = {-\dfrac{3}{2}y - \dfrac{5}{6}}$ Substitute this expression for $x$ in the first equation. $2({-\dfrac{3}{2}y - \dfrac{5}{6}}) + 9y = 1$ $-3y - \dfrac{5}{3} + 9y = 1$ Simplify by combining terms, then solve for $y$ $6y - \dfrac{5}{3} = 1$ $6y = \dfrac{8}{3}$ $y = \dfrac{4}{9}$ Substitute $\dfrac{4}{9}$ for $y$ in the top equation. $2x+9( \dfrac{4}{9}) = 1$ $2x+4 = 1$ $2x = -3$ $x = -\dfrac{3}{2}$ The solution is $\enspace x = -\dfrac{3}{2}, \enspace y = \dfrac{4}{9}$.